### Optimal Query Complexity Bounds for Finding Graphs

#### Jeong Han Kim (National Institute for Mathematical Sciences (NIMS), Korea)

(Joint work with S. Choi)We consider the problem of finding an unknown graph by using two
types of queries with an additive property. Given a graph, an
additive query asks the number of edges in a set of vertices while
a cross-additive query asks the number of edges crossing between
two disjoint sets of vertices. The queries ask sum of weights for
the weighted graphs. These types of queries were partially
motivated in DNA shotgun sequencing and linkage discovery problem
of artificial intelligence.
For a given unknown weighted graph *G* with *n* vertices, *m*
edges, and a certain mild condition on weights, we prove that
there exists a non-adaptive algorithm to find the edges of *G*
using *O((mlogn)/(logm))* queries of both
types provided that *m ≥n ^{ε}* for any constant

*ε> 0*. For a graph, it is shown that the same bound holds for all range of

*m*. This settles a conjecture of Grebinski for finding an unweighte graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions. A similar coin weighing problem is also considered.